Question
Consider RN the product of denumerably many copies of R. Define f: R → RN by...
Consider RN the product of denumerably many copies of R. Define
f: R → RN by f(t) = (t, t,...).
Show that f is continuous if RN has the product
topology, but is not continuous if RN has the box
topology.
Homework Answers
Answer #1
Not the answer you're looking for?
Similar Questions
Define X = {0, 1} and T = {∅, {0}, X} . (a) Is X with...
Define X = {0, 1} and T = {∅, {0}, X} .
(a) Is X with topology T connected? (Hint: Use the clopen
definition.)
(b) Is X with topology T path-connected? (Hint: Construct
continuous map f ∶ R → X. One way is to ensure f −1({0}) = (−∞, 0).
Once you have f, consider f([a, b])—like f([−1, 1]) if you take my
suggestion. Use the definition of path connected. )
Consider two copies of the real numbers: one with the usual topology and one with the...
Consider two copies of the real numbers: one with the usual
topology and one with
the left-hand-side topology. Endow the plane with the product
topology determined by these two
topologies.
Describe and give three examples of the open sets of the
topological subspace { (x,y)| y=x}.
Prove or provide a counterexample Let f:R→R be a function. If f is T_U−T_C continuous, then...
Prove or provide a counterexample
Let f:R→R be a function. If f is T_U−T_C continuous, then f is
T_C−T_U continuous.
T_U is the usual topology and T_C is the open half-line
topology
Let (X1,d1) and (X2,d2) be metric spaces, and let y∈X2. Define f:X1→X2 by f(x) =y for...
Let (X1,d1) and (X2,d2) be metric spaces, and let y∈X2. Define
f:X1→X2 by f(x) =y for all x∈X1. Show that f is continuous.
(TOPOLOGY)
Consider the function f : R → R defined by f(x) = ( 5 + sin...
Consider the function f : R → R defined by f(x) = ( 5 + sin x if
x < 0, x + cos x + 4 if x ≥ 0. Show that the function f is
differentiable for all x ∈ R. Compute the derivative f' . Show that
f ' is continuous at x = 0. Show that f ' is not differentiable at
x = 0. (In this question you may assume that all polynomial and
trigonometric...
Let f : R → R be a continuous function which is periodic. Show that f...
Let f : R → R be a continuous function which is periodic. Show
that f is bounded and has at least one fixed point.
Let f: R --> R be a differentiable function such that f' is bounded. Show that...
Let f: R --> R be a differentiable function such that f' is
bounded. Show that f is uniformly continuous.
Show that f: R -> R, f(x) = x^2 + 2x is not uniformly continuous on...
Show that f: R -> R, f(x) = x^2 + 2x is not uniformly
continuous on R.
Problem 4. Show that if f : R → R is absolutely continuous, then f maps...
Problem 4. Show that if f : R
→ R is absolutely continuous, then f maps sets of
measure zero to sets of measure zero.
Let f : R → R be a function satisfying |f(x) − f(y)| ≤ 3|x −...
Let f : R → R be a function satisfying |f(x) − f(y)| ≤ 3|x −
y|^{1/2} for all x, y ∈ R. Apply E − δ definition to show that f is
uniformly continuous in R.
ADVERTISEMENT
Need Online Homework Help?
Get Answers For Free
Most questions answered within 1 hours.
ADVERTISEMENT
Active Questions
asked 2 minutes ago
asked 16 minutes ago
asked 21 minutes ago
asked 1 hour ago
asked 1 hour ago
asked 1 hour ago
asked 2 hours ago
asked 2 hours ago
asked 2 hours ago
asked 2 hours ago
asked 2 hours ago
asked 2 hours ago